Polynomials with Height 1 and Prescribed Vanishing at 1
Peter Borwein, Michael J. Mossinghoff · Experimental Mathematics · 2000
We study the minimal degree d(m) of a polynomial with all coefficients in {—1, 0, 1} and a zero of order m at 1. We determine dlm) for m ≤ 10 and compute all the extremal polynomials. We also determine the minimal degree for m = 11 and m = 12 among certain symmetric polynomials, and we find explicit examples with small degree for m ≤ 21. Each of the extremal examples is a pure product polynomial. The method uses algebraic number theory and combinatorial computations and relies on showing that a, polynomial with bounded degree, restricted coefficients, and a zero of high order at 1 automatically vanishes at several roots of unity.